Recently I read an interesting article about negative temperature. I was puzzled because I thought before that temperature has definite meaning in thermodynamics: it tells about how fast atoms jiggle. Now if temperature can be negative, this means temperature has wider meaning...

I wonder if there is any deep physical meaning of temperature, not mathematical one.

3 Answers
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where $S$ is the thermodynamic entropy of the system and $U$ its internal energy. The thermodynamic concept of temperature $T$ is more general than the kinetic temperature $T_\mathrm{kin}$ (which only measures the average speed of molecules) because molecules do more than just translate in space.

A negative temperature simply means that the entropy of the system decreases $\delta S < 0$ when you add more energy $\delta U > 0$.

Negative temperatures can only occur in systems where the energy spectrum is bounded above.The systems with Negative-temperature have the opposite characteristics. Adding energy reduces their disorder. But they are not cold in the conventional sense that heat will flow into them from the systems at (+)ve temperatures. In fact, systems with (-)ve absolute temperatures contain more atoms in high-energy states than is possible even at the hottest positive temperatures, so heat should always flow from them to systems above 0K.

That is only true in systems that actually have a translational degree of freedom. The best example for a system without translational freedom (that also exhibits negative temperature) is that of a number of spin-1/2 particles in a line, trapped so that they are unable to move. If put into a magnetic field, the temperature of this system is related to the number of spins "flipped up" vs. spins "flipped down", and has nothing to do with their (non-existent) kinetic energy. juanrga's answer is the right way to define temperature.
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ACuriousMindJul 30 '14 at 12:29